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Using the unique torus decomposition, I will explain how the set of
homeomorphism classes of $3$-manifolds may, in a naive way, be viewed as
a fractal set. Recent advances indicate that much of the complicated
combinatorial and algebraic structure normally associated to graph
manifolds survives even modulo homology cobordism (and exists even in
hyperbolic 3-manifolds!). I will focus on examples that are knot and
link exteriors and discuss an array of invariants starting from
classical signatures and ending with noncommutative algebra and
functional analysis, that have recently been useful in giving evidence
for this extremely complicated fractal behavior. This is joint work with
Shelly Harvey and Constance Leidy.
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